mlsanityData to 7 Oct 2026

Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks

arXiv 2004.12131 ↗First posted 25 Apr 2020math.NA cs.LG cs.NAcited by 27

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Abstract (arXiv)

We perform a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of numerical analysis. As the underlying model, we study the machine-learning-based solution of parametric partial differential equations. Here, approximation theory predicts that the performance of the model should depend only very mildly on the dimension of the parameter space and is determined by the intrinsic dimension of the solution manifold of the parametric partial differential equation. We use various methods to establish comparability between test-cases by minimizing the effect of the choice of test-cases on the optimization and sampling aspects of the learning problem. We find strong support for the hypothesis that approximation-theoretical effects heavily influence the practical behavior of learning problems in numerical analysis.

Key facts. Cited by 27 papers in the map (data through 7 Oct 2026). First posted on arXiv 25 Apr 2020.

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Citing papers per year: 2021: 3; 2022: 6; 2023: 4; 2024: 9; 2025: 5.

27 papers.

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  1. 2022Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activationscited by 29
  2. 2021A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equationscited by 28
  3. 2022Multigrid-augmented deep learning preconditioners for the Helmholtz equationcited by 24
  4. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  5. 2021Partition of unity networks: deep hp-approximationcited by 15
  6. 2023CS4ML: A general framework for active learning with arbitrary data based on Christoffel functionscited by 11
  7. 2024Agnostic Active Learning of Single Index Models with Linear Sample Complexitycited by 11
  8. 2022Born-Infeld (BI) for AI: Energy-Conserving Descent (ECD) for Optimizationcited by 10
  9. 2023Multilevel CNNs for Parametric PDEscited by 9
  10. 2024Learning smooth functions in high dimensions: from sparse polynomials to deep neural networkscited by 7
  11. 2022CAS4DL: Christoffel Adaptive Sampling for function approximation via Deep Learningcited by 6
  12. 2022Solving parametric partial differential equations with deep rectified quadratic unit neural networkscited by 5
15 more
  1. 2025Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programscited by 5
  2. 2021Deep neural network surrogates for non-smooth quantities of interest in shape uncertainty quantificationcited by 4
  3. 2024A Low Rank Neural Representation of Entropy Solutionscited by 3
  4. 2023Operator theory, kernels, and Feedforward Neural Networkscited by 2
  5. 2023Approximating Langevin Monte Carlo with ResNet-like Neural Network architecturescited by 2
  6. 2024Adaptive Multilevel Neural Networks for Parametric PDEs with Error Estimationcited by 2
  7. 2025Noisy PDE Training Requires Bigger PINNscited by 2
  8. 2024Multilevel CNNs for Parametric PDEs based on Adaptive Finite Elementscited by 1
  9. 2024Multiscale Corrections by Continuous Super-Resolutioncited by 1
  10. 2025DPG loss functions for learning parameter-to-solution maps by neural networkscited by 1
  11. 2022The Mathematics of Artificial Intelligencecited by 0
  12. 2024An Efficient Deep Learning Approach for Approximating Parameter-to-Solution Maps of PDEscited by 0
  13. 2024Improving PINNs By Algebraic Inclusion of Boundary and Initial Conditionscited by 0
  14. 2025Prediction of steady states in a marine ecosystem model by a machine learning techniquecited by 0
  15. 2025Learning Low Rank Neural Representations of Hyperbolic Wave Dynamics from Datacited by 0